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Time-Asymptotic Interactions of Boltzmann Shock Layers in the Presence of Boundary

W

EN

-C

HING

L

IEN

& S

HIH

-H

SIEN

Y

U

ABSTRACT. We study the time-asymptotic behavior of the Boltzmann shock layers with a given physical boundary in a half-space. As boundary conditions, we prescribe a Maxwellian at the far field and require a spec- ular reflection at the wallx=0. When the macroscopic velocity at the far field is negative, we prove that if the initial data are suitably chosen, then a solution exists globally in time and tends toward the correspond- ing outgoing Boltzmann shock profile as time goes to infinity. The proof is essentially based on the macro-micro decomposition of solutions and the elementary energy methods.

CONTENTS

1. Introduction. . . 1502

2. Preliminary. . . 1507

2.1. Macro-micro decomposition.. . . 1507

2.2. Boltzmann shock profile.. . . 1510

3. Construction of the Approximate Solution. . . 1512

3.1. The macroscopic and microscopic equations.. . . 1512

3.2. Reference macro-micro decomposition and reference norms.. 1516

3.3. Local existence in time.. . . 1517

4. Basic Estimates. . . 1518

4.1. Properties of the macroscopic variables.. . . 1518

4.2. Matrix representation.. . . 1519

4.3. The eigenvalues for the Navier-Stokes shock profile.. . . 1520

5. Stability Analysis. . . 1521

5.1. Lower order energy estimates.. . . 1524

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5.2. Transversal wave estimates.. . . 1533 5.3. Higher order energy estimates.. . . 1541 App. A. Chapman-Enskog Expansion andNavier-Stokes Shock Profiles. 1549 App. B. Estimates on Collision Operators. . . 1553 Acknowledgment.. . . 1554 References. . . 1554

1. INTRODUCTION

Consider the Boltzmann equation with a given physical boundary in a half space.

We study the plane wave propagation on the half spaceR+ = [0,∞) with the following initial and boundary conditions:

Ft+ ξ1Fx = Q(F, F), (x, t, ξ)∈ R+× R+× R3, (1.1)

x→∞limF (x,0, ξ)= ρ+e−|ξ−u+|2/(2T+) p(2π T+)3 , (1.2)

F (0, t, ξ1, ξ2, ξ3)= F(0, t,−ξ1, ξ2, ξ3), (1.3)

whereF (x, t, ξ)is the density function of the gas at timet0, positionx 0, and with the velocity ξ = (ξ1, ξ2, ξ3). ρ+, u+ = (u+1,0,0), and T+ are the macroscopic density, the macroscopic velocity, and the temperature respectively in the thermal equilibrium at the far field. We assume thatu+1 < 0. Here the specific gas constant is assumed to be one.

We also assume the hard sphere collision:

Q(g, h) Z

R3

Z

S2

h

g(ξ0)h(ξ?0)− g(ξ)h(ξ?)i

C(Ω, ξ − ξ?)ddξ?,

whereC(Ω, ξ − ξ?)≡ |Ω · (ξ − ξ?)|, and

ξ0= ξ + (Ω · (ξ?− ξ))Ω, ξ?0 = ξ?− (Ω · (ξ?− ξ))Ω, Ω ∈ S2. In this paper we are interested in the time-asymptotic behavior of the Boltz- mann shock layers. According to condition (1.2), the incoming flow travels along the positivex-axis and strikes the boundaryx = 0. Due to condition (1.3), the incoming flow will be reflected as a wave of some form. We expect that the outgo- ing flow will tend toward a Boltzmann shock profile as time goes to infinity, which is a solution of the equation

−sψ0+ ξ1ψ0= Q(ψ, ψ).

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Heresis the speed of the shock wave, and the space variable is one dimensional, x∈ R. Motivated by the boundary condition, we first construct the approximate solutionϕof (1.1)–(1.3) by superposing two travelling shock wave solutions mov- ing in the opposite directions with the same speed. By writing the solution of (1.1) asF (x, t, ξ)= ϕ(x, t, ξ) + J(x, t, ξ), we obtain the equation forJas follows:

(1.4) Jt+ ξ1Jx= Q(ϕ + J, ϕ + J) − Q(ϕ, ϕ) − (ϕt+ ξ1ϕx− Q(ϕ, ϕ)).

Our goal is to prove the time-asymptotic convergence of the solution to (1.1)–

(1.3) toward the outgoing Boltzmann shock profile; that is, J(x, t, ξ) decays to zero in a suitable norm ast→ ∞.

There is extensive literature on the initial boundary value problems for the Boltzmann equation initiated by Cercignani. Existence, uniqueness and proper- ties of asymptotic behavior are proved for solutions of the Milne and Kramers problems, which are to solve the linearized Boltzmann equation in a half space x >0. Such studies on stationary solutions have been pursued analytically by [2], [8], [9], [27]. The Milne problem has also been studied by asymptotic expansions for the condensation and evaporation by Sone et al [1], [23].

The shock profiles of the Boltzmann equation are first constructed by [4], where the existence and uniqueness of weak plane shocks are obtained by using a projection method similar to Lyapunov-Schmidt method; however, the positivity property of the Boltzmann shock profile cannot be concluded from this approach.

The time-asymptotic stability of the Maxwellian states has been shown by energy methods based on the Fourier transform and spectral analysis [15,22,24,25].

Furthermore, the time-asymptotic stability and positivity of shock profiles are ob- tained in [18]. An elementary energy method is introduced in [18] based on a macro-micro decomposition of the equation into macroscopic and microscopic components to analyze the time-asymptotic stability of nonlinear waves. The de- composition effectively describes the Boltzmann dynamics so that the methods of analyzing viscous conservation laws can be implemented with small modifications.

The positivity of a Boltzmann shock profile is thus shown by the time-asymptotic approach and the maximal principle for the collision operator.

Applying the macro-micro decomposition introduced in [18], we can regard the present problem as a time-asymptotic stability problem, of which the solution tends toward the nonlinear wave pattern, a superposition of two Boltzmann shock layers. We briefly describe the key steps for solving this problem in the following.

Step 1. We first construct the approximate solutionϕof (1.1)–(1.3). (See Sections2.2and3.1).

The state+, m+,E+)is given by (1.2) atx = ∞. Since (1.3) implies that the macroscopic velocity and momentum are zero atx = 0, we can construct a Boltzmann shock profile with these two given conditions. By solving the Rankine- Hugoniot condition (2.4) and equation (2.3), there existρ0,E0and a wave speed s > 0 such that +, m+,E+)and 0,0,E0)can be connected by a travelling

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wave solutionϕ+(x− st, ξ)on the whole spaceR= (−∞, ∞)satisfying

−sϕ0++ ξ1ϕ0+= Q(ϕ+, ϕ+).

Let+(x−st), m+(x−st),E+(x−st))denote the corresponding macroscopic variables ofϕ+(x− st, ξ). We then constructϕ(x, ξ1, ξ2, ξ3)

ϕ+(−x, −ξ1, ξ2, ξ3). We now choose the approximate solutionϕ(x, t, ξ)to be ϕ(x, t, ξ)≡ ϕ+(x− s(t + t0), ξ)+ ϕ(x+ s(t + t0), ξ)− ρ0ω(ξ; 0, T0), where

ω(ξ; 0, T0)= e−|ξ|2/(2T0) p(2π T0)3.

Hereε ≡ |ρ+− ρ0| 1 and T0is the temperature at x = 0. We chooset0 ε3 large enough to study the time-asymptotic state. It follows from the above construction that

ϕ(x, t, ξ1, ξ2, ξ3)= ϕ(−x, t, −ξ1, ξ2, ξ3).

Step 2. Focus on the following equation and choose the suitable initial state J(x,0, ξ). (See Sections3.1and3.3).

Jt+ ξ1Jx = Q(ϕ + J, ϕ + J) − Q(ϕ, ϕ) − (ϕt+ ξ1ϕx− Q(ϕ, ϕ)).

Let F denote a solution of (1.1)–(1.3), extended to the whole space R by setting

F (x, t, ξ1, ξ2, ξ3)= F(−x, t, −ξ1, ξ2, ξ3), forx <0. We treatF as a perturbation of the approximate solutionϕ. Thus, we write

F (x, t, ξ)= ϕ(x, t, ξ) + J(x, t, ξ).

Therefore,J(x, t, ξ)satisfies the above equation (1.4). We then choose the initial stateJ(x,0, ξ)satisfying

Z

−∞

Z

R3

1 ξi

|ξ|2

 J(x,0, ξ)dξdx=0, fori=1,2,3. (1.5)

Due to the conservation laws for the macroscopic variables, it follows that Z

−∞

Z

R3

1 ξi

|ξ|2

 J(x, t, ξ)dξdx=0, fori=1,2,3.

Such property on macroscopic variables allows one to introduce an anti-derivative variable and the energy method can be applied to study the time-asymptotic sta- bility problem.

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Step 3. Derive the equations for applying the energy method. (See Section 3.1for details.)

Introduce the anti-derivative:

W (x, t, ξ) Zx

−∞J(y, t, ξ)dy.

We obtain from (1.4) (1.6) Wt+ ξ1Wx =

Zx

−∞(Q(ϕ+ J, ϕ + J) − Q(ϕ, ϕ) − E(ϕ))dy, whereE(ϕ)= ϕt+ ξ1ϕx− Q(ϕ, ϕ).

We need to make use of the shock profile of the Navier-Stokes equation. Let uNS and TNS denote the velocity and temperature for the corresponding shock profile of the Navier-Stokes equation constructed by the same conditions imposed onϕat the far field. We denote the corresponding local Maxwellians by

ωtr(x, t, ξ)= e−((ξ1−uNS)22232)/(2TNS) p(2π TNS)3

and the collision invariantsψi(x, t, ξ),i=0, . . . ,4, are as follows:

ψ0(x, t, ξ)=1, ψ1(x, t, ξ)= ξ1p− uNS

TNS

, ψi(x, t, ξ)= pξi

TNS

, i=2,3, ψ4(x, t, ξ)= 1

6

1− uNS)2+ ξ22+ ξ32 TNS

3

! .

We introduce the macroscopic and microscopic variablesW0andW1forW:

W0≡ P0W X4 i=0

 Z

W ψidξ

 ψiωtr, W1≡ P1W≡ W − W0,

whereP0is the projection operator on the space spanned byψiωtr,i=0, . . . ,4 andP1is the orthogonal projectionP1= I − P0. We also decomposeJas

J= J0+ J1, J0≡ P0J, J1≡ P1J.

ApplyingP0to equation (1.6) andP1to equation (1.4) separately, we obtain the desired equations as follows:

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() P0tW0+ P0ξ1P0xW0+ P0ξ1J1=0.

(††) P1tJ0+ P1tJ1+ P1ξ1xJ0+ P1ξ1xJ1− L(J1)= D(J) +N(J)− E(ϕ). It should be noticed that we only need the componentW0andJin the proof, and the local existence in time ofW0andJis shown in Section3.3.

We note that the projectionP0 to equation (1.6) can be regarded as a lin- earization around the Navier-Stokes shock profile in the sense that the physical quantitiesuandT in the background Maxwellian are chosen from the shock pro- file of the Navier-Stokes equation.

Step 4. Apply the elementary energy method to(†)and(††)to prove that the solutionJ(x, t, ξ)decays to zero in thek · kref,Lx(L2ξ) norm ast → ∞. Here, the reference norm is defined by

khkref,Lx(L2ξ)=sup

x∈R

Z

R3

h2(2π T0)3/2e|ξ|2/(2T0)dξ.

(See Section5for details.)

We share several technical difficulties with [18]. For the sake of complete- ness, we address them again. First, we need to apply the transversal wave estimate.

When we apply the theory of hyperbolic conservation laws to the macroscopic part, we recover the 3 families of elementary waves. In the present problem, the compressibility of the Navier-Stokes shock profile can be used in the estimate in- volving the first and third families, but not for the second family which produces transversal terms. Originated from the energy estimate for the stability analysis of a viscous shock profile [10], we refine the transversal wave estimate in the cur- rent situation. Secondly, the Boltzmann equation is nonlinear due to the collision operator. We split the collision operatorQinto the linear partLand the nonlin- ear partN. The negative definiteness ofLyields the decaying of the microscopic component. But we are left with the nonlinear partN, which produces terms like k(1+|ξ|)1/2J1kL2ξ. We therefore introduce the norms:k·kref,Lx(L2ξ),k·kref,Lx,t(L2ξ). (See Section3.2.) It leads us to make the right a priori assumption and establish the higher order energy estimate to resolve the nonlinear terms. Finally, we have error terms caused by the approximate solution and the drifting Maxwellians. The facts that the projectionP0is determined by the physical quantities of the Navier- Stokes shock profile and the local Maxwellians ωtr vary along the same shock profile certainly result in several error terms. When the shock strength is suffi- ciently small, the Boltzmann shock and the Navier-Stokes shock are close enough, which allows us to control all those errors. In addition, Kawashima’s method [13]

is applied to control the density term, which is absent in considering the pertur- bation of a global Maxwellian.

We state the main theorem as follows:

Theorem 1.1. Consider the hard sphere model of equations (1.1)–(1.3). Suppose that the shock strength ofϕis sufficiently small. Under the condition (1.5), there exists

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a constantδ0>0 such that if the initial data are sufficient small:

X

|α|≤4

k∂xαW0kLx,t(L2ξ)+ k∂xαtW0kLx,t(L2ξ)



+ X

|α|≤3

k∂xα{(1+ |ξ|)1/2J1}kref,Lx,t(L2ξ)

+ k∂αxt{(1+ |ξ|)1/2J1}kref,Lx,t(L2ξ)

≤ δ0,

then the Cauchy problem (1.4) has a global solution in time, and the solutionJ(x, t, ξ) decays to zero in thek · kref,Lx(L2ξ)norm ast→ ∞.

That is, the solution F (x, t, ξ)to (1.1)–(1.3) tends toward the outgoing shock profileϕ(x, t, ξ)in thek · kref,Lx(L2ξ)norm ast→ ∞.

Remark 1.2. Although both the strength of shocks and the magnitude of perturbations of the initial data are small, these two parameters are chosen inde- pendently. When the strength of the shock is sufficiently small, the existence of the Boltzmann shock profile can be guaranteed and the shocks of the Boltzmann equation and the Navier-Stokes equation are close enough. Therefore, we can make use of the compressibility of the Navier-Stokes shock profiles. As forδ0, the smallness assumption is to close the higher order estimate due to the nonlinear- ity of the Boltzmann equation. This is also a common short point of the energy method for viscous conservation laws.

This paper is organized as follows. Basic facts about the Boltzmann equation and the shock profiles are summarized in Section 2. In Section3, the approx- imate solution for the present problem is constructed and the related equations are derived. Section4collects several technical estimates developed to analyze the macroscopic and microscopic equations. The stability analysis is finally shown in Section5. The Chapman-Enskog expansion and the Navier-Stokes equations are briefly described in AppendixA. The properties of the collision operator for hard spheres are put in the AppendixB.

2. PRELIMINARY

2.1. Macro-micro decomposition. We first recall several important proper- ties of the Boltzmann equation

Ft+ ξ · ∇xF= Q(F, F), (x, t, ξ)∈ R3× R+× R3.

In the present paper we consider the hard sphere as our model, and the collision operator can be written as follows ([11], [12]):

Q(g, h) Z

R3

Z

S2

h

g(ξ0)h(ξ?0)− g(ξ)h(ξ?)i

C(Ω, ξ − ξ?)ddξ?,

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where

ξ0= ξ + (Ω · (ξ?− ξ))Ω, ξ?0 = ξ?− (Ω · (ξ?− ξ))Ω,

Ω ∈ S2.

The functionC(Ω, ξ − ξ?)for a hard sphere is

C(Ω, ξ − ξ?)≡ |Ω · (ξ − ξ?)|.

The local equilibrium distributions are the distributionsF withQ(F , F )=0, for which the only solutions are the Maxwellians

F (ξ)= ρ0ω(ξ;u0, T0), where

ω(ξ;u0, T0)= e−|ξ−u0|2/(2T0) p(2π T0)3 .

The macroscopic densityρ0, the velocityu0= (u01, u02, u03), and the tempera- ture T0 of the local thermal equilibrium state may vary. In the collision process, mass, momentum, and energy are conserved, i.e., for any distributionsF andG,

Z

R3

Q(F , G)dξ=0, Z

R3

ξiQ(F , G)dξ=0, i=1,2,3, (2.1)

Z

R3|ξ|2Q(F , G)dξ=0.

Set the collision invariantsχi(ξ),i=0, . . . ,4, as follows:

χ0(ξ)=1, χi(ξ)= ξip− u0i

T0 , i=1,2,3, χ4(ξ)= 1

6

|ξ − u0|2 T0 3

! ,

which are normalized with respect to the Maxwellian state:

Z

χiχjω(ξ)dξ= δij, i, j=0, . . . ,4.

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The linearized collision operator

L(h)≡ Q(ω, h) + Q(h, ω) is self-adjoint and non-positive, i.e.,

hLg, hi = hg, Lhi, hLg, gi ≤0.

Hereh, iis the inner product on the spaceL2(R3)with respect to the variableξ: hg, hi ≡

Z ghdξ.

In fact,χiω,i=0, . . . ,4, span the null space ofL. We denote byP0the projection operator on the space spanned by χiω, i = 0, . . . ,4, and by P1 the orthogonal projection: P1= I − P0. F can be decomposed into the macroscopic partF0and the microscopic partF1:

F0= P0F= X4 i=0

 Z χiFdξ

 χiω, F1= P1F= F − F0.

It thus follows that

L(F0)=0 and L(F )= L(F1).

We introduce new norms forF (x, t, ξ), which will be used in the energy estimates:

kFkL2ξ(x, t)≡ hF, Fi1/2, kFkLx,t(L2

ξ) sup

(x,t)∈R×R+

kFkL2

ξ(x, t).

We also note that for hard spheres [5] and Grad’s cutoff potentials [11], Lis the sum of a multiplication operator and a compact operatorK:

Lh(ξ)= −ν(ξ)h(ξ) + K(h)(ξ).

The collision frequencyν(ξ)has a positive lower bound. As a result,Lis negative definite on the microscopic part:

Z

R3

F1L(F1)dξ <−ν0

Z

R3

F12dξ

for some positive constantν0. LetP0ker(L)and its orthogonal complement be denoted byP1. We will write the negative operator restricted to the spaceP1as

L¯≡ L

P1 ≤ −ν0.

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Lemma 2.1. For anygi(x, t, ξ)satisfyingP0gi0,

|hg1, Lg2i| ≤ −1

2{γhg1, Lg1i + γ

1hg2, Lg2i}

for any constantγ >0.

Proof. By the self-adjoint and non-positive property ofL. Remark 2.2. For more properties of the collision operator Q, we refer to AppendixB.

2.2. Boltzmann shock profile. Letϕ(x−st, ξ)be a travelling wave solution of the Boltzmann equation

(2.2) Ft+ ξ1Fx= Q(F, F), (x, t, ξ)∈ R × R+× R3. ϕthus satisfies

(2.3) −sϕ0+ ξ1ϕ0= Q(ϕ, ϕ).

Let(ρ, u, T )denote the macroscopic variables of the travelling wave solution:

ρ(x− st) ≡ Z

R3

ϕ(x− st, ξ)dξ, m(x− st) ≡

Z

R3

ξ1ϕ(x− st, ξ)dξ, u m ρ ,

E(x− st) ≡ Z

R3

|ξ|2

2 ϕ(x− st, ξ)dξ, m2

2ρ + ρT

!

E≡ ρ u2 2 + e

! .

Then the states±, m±,E±)limx→±∞(ρ, m,E)(x)satisfy the Rankine-Hugoniot condition:

s(ρ− ρ+)= m− m+,

s(m− m+)= (um+ p)− (u+m++ p+), (2.4)

s(EE+)= u(E+ p)− u+(E++ p+), and the entropy condition

p− p+>0.

Herep≡ ρRT,R1. For the existence of the shock profileϕ, see [4].

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Denote byϕT(x− st, ξ)the corresponding local thermal equilibrium distri- bution:

ϕT(x− st, ξ) ≡ ρ(x − st)e−((ξ1−u(x−st))22223)/(2T (x−st)) p(2π T (x− st))3 . By direct calculations,ϕTsatisfies the following lemma.

Lemma 2.3.

Z

R3− ϕT)dξ=0, Z

R3

ξi− ui

T − ϕT)dξ=0, i=1,2,3, Z

R3

|ξ − u|2 T 3

!

− ϕT)dξ=0. Here for the shock profileϕof (2.2),u1= u,u2= u3=0.

LetNS, uNS,ENS)(x− st)be the approximate shock profile of the Navier- Stokes equation obtained by the Chapman-Enskog expansion, which connects the same end states±, u±,E±). The corresponding local Maxwellians are denoted by

ωtr(x− st, ξ) = e−((ξ1−uNS)22223)/(2TNS) p(2π TNS)3 ,

ϕtr(x− st, ξ) = ρNSe−((ξ1−uNS)22232)/(2TNS) p(2π TNS)3 .

We consider a weak shockϕwith strengthε ≡ |ρ− ρ+| 1. The rate of the profileϕconverging to the Maxwellian equilibrium states is given in the following theorem.

Theorem 2.4. On the Boltzmann shock profile ϕ(x− st, ξ), there exist C1, C2>1 andC3∈ (0,1)such that

|ϕ(x, ξ) − ϕtr(x, ξ)|

(2.5)

≤ C1ε2e−C3ε|x|ρ(x)e−((ξ1−u(x))22223)/(2C2T (x)) p(T (x))3 ,

|∂kxϕ(x, ξ)|

(2.6)

≤ C1ε1+ke−C3ε|x|ρ(x)e−((ξ1−u(x))22232)/(2C2T (x)) p(T (x))3 ,

k=1, . . . ,10.

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Remark 2.5. The above theorem is proved in [4] and the second inequality is the consequence of the scalings. We also refer readers to Appendix C of [18], where the existence of the Boltzmann shock profile with properties (2.5) and (2.6) has been shown by the weighted energy method.

3. CONSTRUCTION OF THEAPPROXIMATESOLUTION

3.1. The macroscopic and microscopic equations. In this section we con- struct the approximate solution of (1.1)–(1.3) by superposing two travelling wave solutions moving in the opposite directions with the same speed.

The state+, m+,E+)is given by (1.2) atx = ∞. Since (1.3) implies that the macroscopic velocity and momentum are zero at x = 0, we first construct a Boltzmann shock profile with these two given conditions. By solving the Rankine- Hugoniot condition (2.4) and equation (2.3), there existρ0,E0and a wave speed s > 0 such that +, m+,E+)and 0,0,E0)can be connected by a travelling wave solution ϕ+(x− st, ξ) on the whole space R = (−∞, ∞). Let +(x st), m+(x− st),E+(x− st))denote the corresponding macroscopic variables of ϕ+(x− st, ξ). According to the construction,

x→∞lim+(x), m+(x),E+(x))= (ρ+, m+,E+),

x→−∞lim +(x), m+(x),E+(x))= (ρ0,0,E0).

We can also construct the other travelling wave solutionϕ(x+st, ξ)in the same way, satisfying

x→∞lim(x), m(x),E(x))= (ρ0,0,E0),

x→−∞lim (x), m(x),E(x))= (ρ+,−m+,E+).

In fact, we can constructϕ(x, ξ1, ξ2, ξ3)= ϕ+(−x, −ξ1, ξ2, ξ3). It thus follows that

ρ+(x)= ρ(−x), m+(x)= −m(−x),

E+(x)=E(−x), T+(x)= T(−x).

We now choose the approximate solutionϕ(x, t, ξ)to be

(3.1) ϕ(x, t, ξ)≡ ϕ+(x− s(t + t0), ξ)+ ϕ(x+ s(t + t0), ξ)− ρ0ω(ξ; 0, T0).

Heret0≡ ε3,ε≡ |ρ+− ρ0| 1. It follows from the above construction that ϕ(x, t, ξ1, ξ2, ξ3)= ϕ(−x, t, −ξ1, ξ2, ξ3).

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Let F denote a solution of (1.1)–(1.3), extended to the whole space R by setting

F (x, t, ξ1, ξ2, ξ3)= F(−x, t, −ξ1, ξ2, ξ3), forx <0. We now write

F (x, t, ξ)≡ ϕ(x, t, ξ) + J(x, t, ξ).

Then by (1.1) and (3.1),J(x, t, ξ)satisfies

Jt+ ξ1Jx= Q(ϕ + J, ϕ + J) − Q(ϕ, ϕ) − E(ϕ), where

E(ϕ)≡ ϕt+ ξ1ϕx− Q(ϕ, ϕ).

Also,

J(x, t, ξ1, ξ2, ξ3)= J(−x, t, −ξ1, ξ2, ξ3).

We choose the initial stateJ(x,0, ξ)satisfying Z

−∞

Z

R3

1 ξi

|ξ|2

 J(x,0, ξ)dξdx=0, fori=1,2,3.

Due to the conservation laws for the macroscopic variables, it thus follows that Z

−∞

Z

R3

1 ξi

|ξ|2

 J(x, t, ξ)dξdx=0, fori=1,2,3.

We consider the anti-derivative:

W (x, t, ξ) Zx

−∞J(y, t, ξ)dy.

We thus have

Wt+ ξ1Wx= Zx

−∞



Q(ϕ+ J, ϕ + J) − Q(ϕ, ϕ) − E(ϕ)

dy.

Let 

ρNS± , u±NS,E±NS



(x∓ st)

be approximate shock profiles of the Navier-Stokes equation connecting the same end states as those ofϕ±(x∓ st, ξ). Define

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In this paper, we build a new class of neural networks based on the smoothing method for NCP introduced by Haddou and Maheux [18] using some family F of smoothing functions.

Akira Hirakawa, A History of Indian Buddhism: From Śākyamuni to Early Mahāyāna, translated by Paul Groner, Honolulu: University of Hawaii Press, 1990. Dhivan Jones, “The Five

Microphone and 600 ohm line conduits shall be mechanically and electrically connected to receptacle boxes and electrically grounded to the audio system ground point.. Lines in

/** Class invariant: A Person always has a date of birth, and if the Person has a date of death, then the date of death is equal to or later than the date of birth. To be